Very weakly dominated strategy ($u_i(s_i,s_{-i})\le u_i({\sigma }_i,s_{-i})\text{ for every }{s}_{-i}$)

A pure strategy is very weakly dominated by another strategy when its payoff is no greater against every opposing pure strategy profile, with no strict advantage required anywhere. The definition permits payoff-equivalent strategies and even domination by the original strategy itself. A weakly dominated strategy instead has a strict inequality for at least one opposing strategy profile.

In a binary game, suppose ${\sigma }_i$ very weakly dominates $s_i$ and assigns positive probability to a different pure action. At every opposing strategy profile where $s_i$ pays one, all actions in the support of a mixed strategy must also pay one. Where $s_i$ pays zero, their payoffs are automatically no smaller. Therefore every distinct supported action very weakly dominates $s_i$. If the mixed domination is weak with a strict improvement somewhere, choose a supported action paying one at a profile where $s_i$ pays zero; that action also supplies a strict improvement. The requirement of nontrivial support is indispensable for the very-weak version.